Here's a repeat of a post about Jackson's restaurant scene -- if you are looking for something in particular, reply and I'll try to steer you in that direction.
I'm not sure whether you're looking for something "local" or not, but Nick's has the best food in town IMO. It's fine dining. Walker's is also great -- it's our neighborhood place and where I eat more than any other. It's a little more casual. Plus you can eat out on the patio if the weather is nice. Pavoni's is a new place that's very good -- I posted a review after our dinner there New Year's Eve. It's brown bag. Parker House is another fine dining restaurant that can be very good. DW LOVES Bon Ami; I like it as well but not quite as much as she does. Also brown bag. Tico's is dreadfully tacky -- and the sides are barely passable -- but they have the best steak in town. Amerigo's is probably the best Italian place in town, but I don't find myself there very frequently. Bravo is good, inventive food, and they have a decent wine list. AJ's is a good seafood place, but a couple of people who ate there recently didn't have good things to say so it may have gone downhill. The Mayflower downtown is an institution; they cook a good piece of fish and you can bring in wine. Julep's can be good, but it is really inconsistent. It's probably the most "Southern" place on this list, though. If you're looking for a more authentic Southern experience, let me know, and I'll post some suggestions. It's just that type of food isn't my favorite. Heresy, I know ... In sum, the food at Nick's is great, but I prefer the atmosphere at Walker's. I've been to Pavoni's several times recently and loved it each time. It is reservation only (no walk-ins) and it can take a while, so if you are not OK with that try somewhere else.